Signal Analysis Music 270a: Signal Analysis Tamara Smyth, [email protected] • Some tools we may want to use to automate analysis Department of Music, are: University of California, San Diego (UCSD) 1. Amplitude Envelope Follower December 2, 2019 2. Peak Detection (attacks or harmonics), surfboard method 3. Pitch Detection, harmonics vs. chaotic signal 4. Frequency/Spectral Envelope (formant tracking, mccs or lpc) 5. Constant overlap-add (COLA) 1 Music 270a: Signal Analysis 2 Amplitude Envelope Follower Peak Detection • An envelope follower will essentially determine the • It is often desirable to automatically detect peaks, amplitude envelope without dipping down into the particularly in a spectral envelope. valleys/zero crossings. • The peak is the location where the slope changes • It is a reduction of the information, representing the directions. overal shape of the signal’s amplitude (i.e. amplidtude spectrum magnitude (linear) envelpe) without the higher frequency information. 14 12 • The amplitude envelope y(n) is given by 10 8 6 y(n)=(1 − ν)|x(n)| + νy(n − 1), 4 magnitude (linear) 2 0 where ν determines how quickly changes in x(n) are 0 500 1000 1500 2000 2500 3000 3500 4000 tracked: magnitude zoomed on first peak 14 – 12 if ν is close to one, changes are tracked slowly 10 – 8 if ν is close to zero, x(n) has an immediate 6 influence on y(n). 4 magnitude (linear) 2 0 600 620 640 660 680 700 720 740 760 780 800 • In order to capture attacks in the signal, the value for frequency (Hz) ν is usually smaller for an increasing signal and large Figure 1: Peak Detection for one that is decreasing. • See envfollower.m % threshhold Music 270a: Signal Analysis 3 Music 270a: Signal Analysis 4 th = 0; Quadratic interpolation % filter descending values uslope = Ymag > [Ymag(1); Ymag(1:end-1)]; % filter ascending values dslope = Ymag >= [Ymag(2:end); 1+Ymag(end)]; • The position of the peak is limited by the resolution % only indeces at maxima retain non-zero value Ymax = Ymag .* (Ymag > th) .* uslope .* dslope; of the DFT/FFT and its estimation can be improved using quadratic interpolation. % peak indeces maxixs = find(Ymax); • The general equation for a parabola is given by y(x) , a(x − p)2 + b, where p is the peak location and b = y(p). • Considering the parabola at points x =0, 1, −1 yields 3 equations, y(0) = ap2 + b = β y(1) = a(1 + p2 − 2p)+ b = γ y(−1) = a(1 + p2 +2p)+ b = α, and 3 unknowns a,p,b. • Solve for a: α − γ α − γ =4ap −→ a = . 4p Music 270a: Signal Analysis 5 Music 270a: Signal Analysis 6 • Solve for p using a: Pitch Detection γ − β = a − 2ap α − γ α − γ = − 2 p 4p 4p • Considerations for Computer Music applications: 4p(γ − β) = α − γ − 2(α − γ)p 1. Signals are often noisy, eg: poor soundcards, other 4p(γ − β)+2p(α − γ) = α − γ instruments/voices, 2p(γ − 2β + α) = α − γ α − γ 2. How much frequency resolution is needed? Correct p = . octave a must, but will a semitone suffice? 2(γ − 2β + α) 3. What latency can be tolerated (what framesize should be used for analysis?) • Finally, the height at peak p is given by 4. Does the instrument have well-defined/behaved y(p) = b harmonics? = β − ap2 α − γ • In the paper by de la Cuadra et al., “Efficient Pitch = β − p2. 4p Detection Techniques for Interactive Music”, four (4) pitch detection algorithms are summarized: • Another way (Dan Ellis) 1. Harmonic Product Spectrum 2. Maximum Likelihood 3. Cepstrum-Biased HPS 4. Weighted Autocorrelation Function Music 270a: Signal Analysis 7 Music 270a: Signal Analysis 8 Harmonic Product Spectrum (HPS) • Due to noise, frequencies below about 50 Hz should not be searched for a pitch. • Pros: HPS is simple to implement, does well under a • HPS (Noll 1969) measures the maximum coincidence wide range of conditions, and runs in real-time. for harmonics for each spectral frame according to • Cons: low frequency resolution must be enhanced by R zero-padding, so that the spectrum can be Y (ω)= Y |X(ωr)|, (1) interpolated to the nearest semitone. This means that r=1 high frequencies are also being unecessarily where R is the number of harmonics being considered. interpolated. • The resulting periodic correlation array Y (ω) is then • See hps.m searched for a maximum value of a range of possible fundamental frequencies ωi X(ω ) 0 Yˆ = max Y (ωi) (2) ωi X(ω ) 1 to obtain the fundamental frequency estimate. X(ω ) • Octave errors are common (detection is sometimes an 2 octave too high). X(ω ) 3 • To correct, apply this rule: if the second peak X(ω ) amplitude below initially chosent pitch is 4 approximately 1/2 of the chosen pitch AND the ratio of amplitudes is above a threshold (e.g., 0.2 for 5 Figure 2: Harmonic Product Spectrum harmonics), THEN select the lower octave peak as the pitch for the current frame. Music 270a: Signal Analysis 9 Music 270a: Signal Analysis 10 Uses of Linear Predictive Coding Concept of LPC (LPC) • Given a digital system, can the value of any sample • LPC, a statistical method for predicting future values be predicted by taking a linear combination of the of a waveform on the basis of its past values1, is often previous N samples? used to obtain a spectral envelope. • Stated mathematically, can a set of coefficients, ak, • LPC differs from formant tracking in that: be determined such that – the waveform remains in the time domain; y(n)= a1y(n − 1)+ a2y(n − 2)+ ... + aN y(n − N). resonances are described by the coefficients of an all-pole filter. • That is, can a signal be represented as coefficients of an all-pole (only feedback terms) IIR filter. – altering resonances is difficult since editing IIR filter coefficients can result in an unstable filter. • If yes, then the coefficients and the first N samples – analysis may be applied to a wide range of sounds. would completely determine the remainder of the signal, because the rest of the samples can be • LPC is often used to determine the filter in a calculated by the above equation. source-filter model of speech2 which: – characterizes the response of the vocal tract. – reconstitutes the speech waveform when driven by the correct source. 1Markel, J. D., and Gray, A. H., Hr. Linear Prediction of Speech. New York: Apringer-Verlag, 1976. 2In a source-filter model of speech, there is assumed to be no feedback dependency between the vibrating vocal folds and the vocal track Music 270a: Signal Analysis 11 Music 270a: Signal Analysis 12 LPC residual LPC Order • The answer is actually “not precisely” for a finite N. • The accuracy of the predictor improves with an • Rather, we determine the cofficients that give the increase of order N: best prediction by minimizing the difference, or error • The smallest value of N that will yield sufficiently e(n), between the actual sample values of the input quality in the speech representation is related to waveform y(n) and the waveform re-created using the 1. the highest frequency in the speech, derived predictors yˆ(n). 2. the number of formant peaks expected and min{e(n)} = min{yˆ(n) − y(n)}. n n 3. the sampling rate. • The smaller the average value of the error, also called • There is no exact relationship for determining what the residual, the better the set of predictors. value of N should be used, but in general it’s between • The residual may be used to exactly reconstruct the 10 and 20 or the sampling rate in kHz plus 4 (for original signal y(n) by using it as an input to our fs = 15kHz, you might use a N = 19). all-pole filter, that is • Schemes for determining predictors by minimizing the y(n)= b0e(n)+a1y(n−1)+a2y(n−2)+...+aNy(n−N) residual work either explicitly or implicitly. where b0 is a scaling factor that gives the correct • A more rigorous treatment of the subject can be amplitude. found in J. Makhoul’s tutorial3 • When the speech is voiced, the residual is essentially a periodic pulse waveform with the same fundamental frequency as the speech. When unvoiced, the residual 3Makhoul, J. “Linear Prediction, a Tutorial Review.” Proceedinds of the institute of Electrical and is similar to white noise. Electronics Engineers, 63, 1975, 561-580. Music 270a: Signal Analysis 13 Music 270a: Signal Analysis 14 LPC Analysis in Matlab • Matlab for generating original, lpc, and residual spectra. LPC Analysis of the vowel "aah" 25 %lpc original spectrum lpc spectrum order = fs/1000 + 5; % order residual spectrum xlpc = lpc(xw, order)’; % coefficients 20 %windowed speech frequency response 15 zpf = 3; Nfft = 2^nextpow2(N*zpf); XW = fft(xw, Nfft); 10 % lpc spectrum [H,w] = freqz(1, xlpc, Nfft, ’whole’); 5 0 %inverse filter to obtain residual 0 500 1000 1500 2000 2500 E = XW./H; Figure 3: LPC Analysis Results for vowel ’a’. e = real(ifft(E)); Music 270a: Signal Analysis 15 Music 270a: Signal Analysis 16 Constant Overlap Add (COLA) signal X(ω), that is: ∞ ∞ ∞ , −jωn X Xm(ω) X X x(n)w(n − mR)e m=−∞ m=−∞ n=−∞ • Mathematical definition of the Short-time Fourier ∞ ∞ −jωn Transform (STFT) is given by = X x(n)e X w(n − mR) ∞ n=−∞ m=−∞ X ω x n w n mR e−jωn, ∞ m( )= X ( ) ( − ) −jωn n=−∞ = X x(n)e = X(ω), if COLA. n=−∞ where R is the hopsize, and m is the length of the window.